Search for Double-Beta Decay of \mathrm{^{130}Te} to the 0^+ States of \mathrm{^{130}Xe} with CUORE
CUORE Collaboration, D. Q. Adams, C. Alduino, K. Alfonso, F. T. Avignone III, O. Azzolini, G. Bari, F. Bellini, G. Benato, M. Biassoni, A. Branca, C. Brofferio, C. Bucci, J. Camilleri, A. Caminata, A. Campani, L. Canonica, X. G. Cao, S. Capelli, L. Cappelli, L. Cardani, P. Carniti, N. Casali, E. Celi, D. Chiesa, M. Clemenza, S. Copello, C. Cosmelli, O. Cremonesi, R. J. Creswick, A. D'Addabbo, I. Dafinei, C. J. Davis, S. Dell'Oro, S. Di Domizio, V. Dompè, D. Q. Fang, G. Fantini, M. Faverzani, E. Ferri, F. Ferroni, E. Fiorini, M. A. Franceschi, S. J. Freedman, S. H. Fu, B. K. Fujikawa, A. Giachero, L. Gironi, A. Giuliani, P. Gorla, C. Gotti, T. D. Gutierrez, K. Han, K. M. Heeger, R. G. Huang, H. Z. Huang, J. Johnston, G. Keppel, Yu. G. Kolomensky, C. Ligi, L. Ma, Y. G. Ma, L. Marini, R. H. Maruyama, D. Mayer, Y. Mei, N. Moggi, S. Morganti, T. Napolitano, M. Nastasi, J. Nikkel, C. Nones, E. B. Norman, A. Nucciotti, I. Nutini, T. O'Donnell, J. L. Ouellet, S. Pagan, C. E. Pagliarone, L. Pagnanini, M. Pallavicini, L. Pattavina, M. Pavan, G. Pessina, V. Pettinacci, C. Pira, S. Pirro, S. Pozzi, E. Previtali, A. Puiu, C. Rosenfeld, C. Rusconi, M. Sakai, S. Sangiorgio, B. Schmidt, N. D. Scielzo, V. Sharma, V. Singh, M. Sisti, D. Speller, et al. (14 additional authors not shown)
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Search for Double-Beta Decay of Te to the + States of Xe withCUORE D. Q. Adams , C. Alduino , K. Alfonso , F. T. Avignone III , O. Azzolini , G. Bari ,F. Bellini , G. Benato , M. Biassoni , A. Branca , C. Brofferio , C. Bucci ,J. Camilleri , A. Caminata , A. Campani , L. Canonica , X. G. Cao ,S. Capelli , L. Cappelli , L. Cardani , P. Carniti , N. Casali , E. Celi ,D. Chiesa , M. Clemenza , S. Copello , C. Cosmelli , O. Cremonesi ,R. J. Creswick , A. D’Addabbo , I. Dafinei , C. J. Davis , S. Dell’Oro ,S. Di Domizio , V. Dompè , D. Q. Fang , G. Fantini , M. Faverzani , E. Ferri ,F. Ferroni , E. Fiorini , M. A. Franceschi , S. J. Freedman , S.H. Fu ,B. K. Fujikawa , A. Giachero , L. Gironi , A. Giuliani , P. Gorla , C. Gotti ,T. D. Gutierrez , K. Han , K. M. Heeger , R. G. Huang , H. Z. Huang , J. Johnston ,G. Keppel , Yu. G. Kolomensky , C. Ligi , L. Ma , Y. G. Ma , L. Marini ,R. H. Maruyama , D. Mayer , Y. Mei , N. Moggi , S. Morganti , T. Napolitano ,M. Nastasi , J. Nikkel , C. Nones , E. B. Norman , A. Nucciotti , I. Nutini ,T. O’Donnell , J. L. Ouellet , S. Pagan , C. E. Pagliarone , L. Pagnanini ,M. Pallavicini , L. Pattavina , M. Pavan , G. Pessina , V. Pettinacci , C. Pira ,S. Pirro , S. Pozzi , E. Previtali , A. Puiu , C. Rosenfeld , C. Rusconi ,M. Sakai , S. Sangiorgio , B. Schmidt , N. D. Scielzo , V. Sharma , V. Singh ,M. Sisti , D. Speller , P.T. Surukuchi , L. Taffarello , F. Terranova , C. Tomei ,K. J. Vetter , M. Vignati , S. L. Wagaarachchi , B. S. Wang , B. Welliver ,J. Wilson , K. Wilson , L. A. Winslow , S. Zimmermann , S. Zucchelli Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA INFN – Laboratori Nazionali di Legnaro, Legnaro (Padova) I-35020, Italy INFN – Sezione di Bologna, Bologna I-40127, Italy Dipartimento di Fisica, Sapienza Università di Roma, Roma I-00185, Italy INFN – Sezione di Roma, Roma I-00185, Italy INFN – Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila) I-67100, Italy INFN – Sezione di Milano Bicocca, Milano I-20126, Italy Dipartimento di Fisica, Università di Milano-Bicocca, Milano I-20126, Italy Center for Neutrino Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA INFN – Sezione di Genova, Genova I-16146, Italy Dipartimento di Fisica, Università di Genova, Genova I-16146, Italy Massachusetts Institute of Technology, Cambridge, MA 02139, USA Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Department of Physics, University of California, Berkeley, CA 94720, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA INFN – Gran Sasso Science Institute, L’Aquila I-67100, Italy Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06520, USA INFN – Laboratori Nazionali di Frascati, Frascati (Roma) I-00044, Italy Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Physics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA INPAC and School of Physics and Astronomy, Shanghai Jiao Tong University; Shanghai Laboratory for Particle Physics and Cosmology,Shanghai 200240, China Dipartimento di Fisica e Astronomia, Alma Mater Studiorum – Università di Bologna, Bologna I-40127, Italy Service de Physique des Particules, CEA / Saclay, 91191 Gif-sur-Yvette, France Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Department of Nuclear Engineering, University of California, Berkeley, CA 94720, USA Dipartimento di Ingegneria Civile e Meccanica, Università degli Studi di Cassino e del Lazio Meridionale, Cassino I-03043, Italy Department of Physics and Astronomy, The Johns Hopkins University, 3400 North Charles Street Baltimore, MD, 21211 INFN – Sezione di Padova, Padova I-35131, Italy Engineering Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAReceived: date / Accepted: date a r X i v : . [ nu c l - e x ] J a n Abstract
The CUORE experiment is a large bolometric ar-ray searching for the lepton number violating neutrino-lessdouble beta decay (0 νβ β ) in the isotope
Te. In this workwe present the latest results on two searches for the dou-ble beta decay (DBD) of
Te to the first 0 + excited stateof Xe: the 0 νβ β decay and the Standard Model-allowedtwo-neutrinos double beta decay (2 νβ β ). Both searches arebased on a 372.5 kg × yr TeO exposure. The de-excitationgamma rays emitted by the excited Xe nucleus in the finalstate yield a unique signature, which can be searched forwith low background by studying coincident events in twoor more bolometers. The closely packed arrangement of theCUORE crystals constitutes a significant advantage in thisregard. The median limit setting sensitivities at 90% Credi-ble Interval (C.I.) of the given searches wereestimated as S ν / = . × yr for the 0 νβ β decay andS ν / = . × yr for the 2 νβ β decay. No significant ev-idence for either of the decay modes was observed and aBayesian lower bound at 90% C.I. on the decay half livesis obtained as: ( T / ) ν + > . × yr for the 0 νβ β modeand ( T / ) ν + > . × yr for the 2 νβ β mode. These rep-resent the most stringent limits on the DBD of Te to ex-cited states and improve by a factor ∼ Double beta decay (DBD) is an extremely rare nuclear pro-cess where a simultaneous transmutation of a pair of neu-trons into protons converts a nucleus (A, Z) into an isobar(A, Z+2), with the emission of two electrons and two anti-neutrinos. This two-neutrino decay mode (2 νβ β ) is pre-dicted in the Standard Model and was detected in several nu-clei. The neutrinoless mode of the decay (0 νβ β ) is a positedBeyond Standard Model process that could shed light onmany open aspects of modern particle physics and cosmol-ogy such as the existence of lepton number violation and el-ementary Majorana fermions, the neutrino mass scale, andthe baryon asymmetry in the Universe [1–5]. Both DBDmodes can proceed through transitions to the ground stateas well as to various excited states of the daughter nucleus.While the former can be easier to detect through their shorterhalf-lives, the latter leaves a unique signature which maybe detected with significantly reduced backgrounds. The ex-cited state decays also provide powerful tests of the nuclearphysics of DBD and can shed light on nuclear matrix ele-ment calculations as well as the ongoing discussion on thequenching of the effective axial coupling constant g A ; even-tually, they could even be used to disentangle the mechanismof 0 νβ β decay [6]. a Deceased
So far, 2 νβ β decay to the first 0 + excited state has beenobserved in only 2 isotopes: Mo [7] and
Nd [8], withhalf lives of ( T / ) ν + = . + . − . × yr and ( T / ) ν + = . + . − . ( stat . ) ± . ( syst . ) × yr, respectively. Searchesfor the same process in other isotopes has yielded lower lim-its from 3 . × yr to 8 . × yr at 90 % ConfidenceLevel (C.L.) (see Ref. [9] for a review).In this work, we focus on the search for 0 νβ β and 2 νβ β decays of Te to the first 0 + excited state of Xe withthe CUORE experiment. Presently, the strongest limits onthe decay to excited states half-life of
Te come from acombination of Cuoricino [10] and CUORE-0 [11] data: thelatter (not included in Ref. [9]) was published recently andincludes the combination of the predecessor’s results. Theobtained limits are: ( T / ) ν + > . × yr (
90% C . L . ) (1a) ( T / ) ν + > . × yr (
90% C . L . ) (1b)for the 0 ν and 2 ν process, respectively. Theoretical predic-tions [12] on the ( T / ) ν + observable in the 2 νβ β decaychannel are based on the QRPA approach and favor the fol-lowing range: th ( T / ) ν + = ( . − ) × yr (2)where the range depends on the precise treatment of g A . Thelower bound assumes a constant function of the mass num-ber A, and the upper bound assumes a value of g A = . νβ β ground state to ex-cited state decay rate in the IBM-II framework based on Ref.[15–17] is reported in Ref. [18] as th ( T / ) ν + = . × yr . (3)In this regard, as stated before, both a measurement or amore stringent limit with respect to Ref. [11] are informa-tive from the point of view of refining and validating thetheoretical computations.The decay to excited states has a unique signature. Thedouble-beta decay emits two electrons, which share kineticenergy up to 734 keV. The subsequent decay of the exciteddaughter nucleus typically emits two or three high energygamma rays in cascade. Due to the emission of such co-incident de-excitation γ rays, both 0 νβ β and 2 νβ β decaychannels allow a significant background reduction with re-spect to the corresponding transitions to the ground state.This holds especially in an experimental setup that exploitsa high detector granularity, such as the CUORE experiment. The Cryogenic Underground Observatory for Rare Events(CUORE) [19, 20] is a ton-scale cryogenic detector locatedat the underground Laboratori Nazionali del Gran Sasso(LNGS) in Italy. CUORE is designed to search for the 0 νβ β decay of
Te to the ground state of
Xe [21, 22], and hasa low background rate near the 0 νβ β decay region of in-terest (ROI), an excellent energy resolution, and a high de-tection efficiency. The CUORE detector consists of a close-packed array of 988 TeO crystals operating as cryogenicbolometers [23–25] at a temperature of ∼
10 mK.The CUORE crystals are 5 × × cubes weighing 750g each, arranged in 19 towers: each consisting of a copperstructure with 13 floors and 4 crystals per floor. A custom-made He/ He dilution refrigerator, which represents thestate of the art for this cryogenic technique, is used to cooldown the CUORE cryostat, where the entire array is con-tained and shielded. [26–32].Each CUORE crystal records thermal pulses via aneutron-transmutation doped (NTD) germanium thermistor[33] glued to its surface. Any energy deposition in the crys-tal causes a sudden rise in temperature and can indicate theemission of a particle inside, the crossing of a particlethrough, or some environmental thermal instability (e.g.earthquakes).The data acquisition and production of CUORE eventdata used in this work closely follows the procedure used in[26] and is described in detail in [34]. We briefly review thebasic process here and highlight the differences.The NTD converts the thermal signal to a voltage out-put, which is amplified, filtered through a 6-pole Bessel anti-aliasing filter, and sampled continuously at 1kHz. The dataare stored to disk and triggered offline with an algorithmbased on the optimum filter (OF) [35–37].For each triggered pulse, a 10 second window aroundeach trigger (3 seconds before and 7 seconds after) is pro-cessed through a series of analysis steps, with the aim of ex-tracting the physical quantities associated to the pulse. Thewaveform is filtered using an OF built from a pulse template,and the measured noise power spectrum.The signal amplitudes are then evaluated from the OFfiltered waveforms and those amplitudes are corrected forsmall changes in the detector gain due to temperature drifts.We calibrate each bolometer individually using dedicatedcalibration runs with
Th and Co gamma sources de-ployed around the detector array. These calibration runs typ-ically last a few days every two months.We impose a pulse shape selection (PSA) based on 6pulse shape parameters. This cut removes noisy events,pileup events, and non-physical events.Unlike the decay to ground state search described in Ref.[26], the physics search described in the present work fo- cuses on coincident energy depositions in multiple crystals.In particular, we are focusing on events where energy isdeposited in either two or three bolometers. As the recon-structed time difference between events on nearby bolome-ters is affected by differences in pulse rise times, a bolometer-by-bolometer correction is applied. Sets of coincident en-ergy releases in M bolometers within a ± M .CUORE started its data taking in May 2017 and, aftertwo significant interruptions for important maintenance ofthe cryogenic system, is now seeing its exposure grow atan average rate of ∼
50 kg × yr/month. The CUORE datacollection is organized in datasets: a dataset begins with agamma calibration campaign that typically lasts 2-3 days,followed by 6-8 weeks of uninterrupted background datataking, and ends with another gamma calibration.Recently, the CUORE collaboration released the resultsof the search for 0 νβ β decay to the g.s. on the the accumu-lated exposure of 372.5 kg · y, setting an improved limit onthe half-life of Te of ( T / ) ν + > . × yr [22]. In this section we describe the analysis steps that are specificto the search for
Te decay to the excited states of
Xe.The de-excitation of the
Xe nucleus follows one of threepossible patterns , i.e. paths through states of decreasing en-ergy from the 0 + to the 0 + ground state (Figure 1). Detailsabout the probability of each de-excitation pattern, referredin the following as A, B and C (in decreasing order of prob-ability), and the energy of the emitted γ rays are reported inTable 1.The simultaneous emission of DBD betas andde-excitation gammas produces coincidence multiplets, i.e.sets of simultaneous pulses in M bolometers, grouped bythe coincidence algorithm. We search for events with fullcontainment of the final state gammas in the crystals: morespecifically we try to avoid multiplets where one or more ofthe final state γ s escape the source crystal and are absorbedby some non-active part of the experimental apparatus, orCompton scattering events, where the energy of a single de-excitation gamma is split among two or more detectors. Weplace energy selection cuts to find these events, which arelisted in Table 3 and described in more details in Sec. 3.4. Partitions are defined as unique groupings of energy de-positions that pass a particular set of energy selection cuts.For a fixed multiplicity M and a source pattern, they areidentified by all possible ways of partitioning the final stateparticles in M different crystals. Finally we define signa-tures as partitions from different patterns that are indistin-guishable. Single-site ( M =
1) signatures are not taken intoaccount, as the 0 νβ β decay channel would be indistinguish- +1 +2 +2 +1 Te I Xe + +
86 14 10013 87 ββ β
Fig. 1
The decay scheme of
Te is shown with details about theinvolved excited states of
Xe up to its first 0 + excited state. Thenomenclature 0 + ,..., + n indicates states with the same angular momen-tum in increasing order of excitation energy. An energy scale is shown(right) where the Xe ground state is taken as reference [39].
Pattern BR [%] Energy γ Energy γ Energy γ A 86% 1257 keV 536 keV -B 12% 671 keV 586 keV 536 keVC 2% 1122 keV 671 keV -
Table 1
The de-excitation γ rays emitted by Xe ∗ in the transitionfrom the 0 + to the ground state. Each row corresponds to a differentpath through intermediate states. The energies of the emitted γ s arelisted, in order of energy, along with the branching ratio (BR) of eachpattern [39]. able from the same decay on the ground state of Xe, whilethe 2 νβ β decay channel, instead, would suffer from highbackground from the decay to the ground state. Therefore,there remain 8 partitions for patterns A and B, and 14 forpattern C. Each of the partitions is labelled with strings of 3characters with the following convention[
Multiplicity ] [
Pattern ] [
Index ]where
Multiplicity = 2,3,4 indicates the number of involvedcrystals,
Pattern = A,B,C stores the originating de-excitationpattern, and
Index is a unique integer counter to distinguishthe various combinations of energy groupings for that pat-tern and multiplicity. Partitions sharing the same expectedenergy release are indistinguishable and are merged as sig-natures. For this reason, instead of handling a total numberof 22 partitions we are left with 15 signatures [38]. An exam-ple of indistinguishable partitions is given in Table 3 by the2A0-2B1 signature. In one of the crystals a gamma energyrelease of 1257 keV is expected. This can be either due (seeTable 1) to γ from pattern A or the simultaneous absorptionof γ + γ from pattern B.3.1 Monte Carlo SimulationsWe use Monte Carlo (MC) simulations to compute the de-tection efficiency (Sec. 3.2) and the expected background (Sec. 3.3) associated with each signature, to rank experimen-tal signatures and eventually fine tune selection cuts on themost sensitive ones (Sec. 3.4).CUORE uses a Geant4-based MC to simulate energy de-positions in the detector. The Geant4 software [40] simulatesparticle interactions in the various volumes and materialsof a modeled detector geometry. A separate post-processingstep converts the resulting energy depositions into an outputas close to the output of the data production as possible. Werefer to Ref. [41] for further details about the CUORE MCsimulations.Signal simulations, that are simulations of the doublebeta decay to excited states and the subsequent de-excitationgammas, are produced separately for each process (0 ν ,2 ν )and pattern (A,B,C). Gamma energies are generated asmonochromatic, angular correlations are not taken into ac-count, beta energies are randomly extracted from the betaspectrum of the corresponding decay [42–44] in the HSDhypothesis . We note though that this analysis is not sen-sitive to the shape of the beta spectrum because in 2 νβ β signatures the fit is performed just on the gamma peaks.Background simulations take as input the CUORE back-ground model [45], and include contaminations in the crys-tal and several other parts of the CUORE setup , such as:the copper tower structure, the closest copper vessel enclos-ing the detector, the Roman lead, the internal and externalmodern lead shields and the internal lead suspension sys-tem. The contaminants include bulk and surface U and
Th chains with different hypotheses on secular equilib-rium breaks, bulk Co, K, and a few other long livedisotopes. Additional sources of background included are thecosmic muon flux and the 2 νβ β decay of
Te to the groundstate. Both signal and background simulated energy spectraare convolved with a Gaussian resolution that has a width of5 keV full width at half maximum, a standard choice for oursimulation studies [41].3.2 Efficiency EvaluationThe detection efficiency of a given signature consists of twocomponents: the containment efficiency and the analysis ef-ficiency. Given a signature s and a set of energy selectioncuts on the involved bolometers, the corresponding contain-ment efficiency ε s represents the probability that the energyreleased by a nuclear decay of Te to the 0 + state of Xe The 2 νβ β is called single state dominated (SSD) if it is governed bythe lowest 1 + energy level. In the higher-state dominated (HSD) casethe calculation is simplified by summing over all the virtual intermedi-ate states and assuming an average closure energy. The CUORE background model we refer to is still preliminary, how-ever the estimates of the background activities are good enough to un-derstand what will be the expected contribution to the present search.In the final fit the exact values are floated. matches the topology of the signature. We evaluate this effi-ciency component from the signal MC simulations describedin Sec. 3.1, by summing over the contributions of all patterns p populating the signature s ε s = (cid:34) ∑ p BR p · (cid:2) N ( sel ) MC (cid:3) ( s ) p (cid:2) N ( tot ) MC (cid:3) p (cid:35) (4)where BR p is the branching ratio of pattern p , (cid:2) N ( sel ) MC (cid:3) ( s ) p and (cid:2) N ( tot ) MC (cid:3) p are respectively the selected and total numberof simulated decays in the de-excitation pattern of interest.For 0 νβ β decay signatures the signal is monochromatic inall the involved crystals, so the signal region is expected tolie around a specific point in the M-dimensional space ofcoincident energy releases. A selection is enforced, in sim-ulations, with a box cut , i.e. a selection interval for energyreleases in each crystal, defined as | E i − Q i | < i = ... M (5)where E i is the reconstructed energy release in the orderedenergy space and Q i is the corresponding expected energyrelease. For 2 νβ β decay signatures the same selections ap-ply except the one crystal where the energy release from the β β is expected. Since the emitted neutrinos carry away anunknown (on an event basis) amount of undetected energy,the expected energy release is not monochromatic. It is in-stead expected to vary from Q minj to Q maxj where j indicatesthe bolometer where the β β release their energy. For thatbolometer, in each multiplet the following selection is ap-plied Q minj − < E j < Q maxj + ε cut ), and the probability of as-signing the correct multiplicity and avoiding an accidentalcoincidence (accidentals efficiency, ε acc ).The cut efficiency term is named after the data process-ing cuts needed to select triggered events that pass the baseand PSA cuts (see Sec. 2). The method used to calculatethis efficiency follows closely what was used in [22]. Wemeasure the efficiency of correctly triggering, reconstruct-ing the pulse energy and the pile-up contribution (base cuts) The energy releases of each M -bolometers multiplet are ordered in de-scending order so that E i > E i + . from heater pulses. The base cut efficiency is computed oneach bolometer-dataset pair given the large number of avail-able heater events, and then averaged to obtain a per-datasetvalue. The PSA cut efficiency is extracted from two indepen-dent samples of events: either coincident double-site eventswhere the total energy released is compatible with knownprominent γ lines, or single-site events due to fully absorbed γ lines. The first sample includes events whose energy spansa wide range, and allows the determination of the PSA cutefficiency dependence on energy. The second sample has ahigher statistics but provides a measurement at the energiesof the selected γ peaks only, rather than on a continuum.We evaluate for each dataset the PSA cut efficiency termas the average of the two efficiencies obtained from suchsamples. We treat the difference as a systematic effect. The ε cut term must be raised to the M th power because it modelsbolometer-related efficiencies and a multiplet is selected ifand only if all of the involved bolometers pass the selectioncuts.The accidentals efficiency term ε acc is obtained, sepa-rately for each dataset, as the survival probability of the K γ line at 1460 keV. A fully absorbed K line in CUOREis uncorrelated to any other physical event because it fol-lows an electron capture of a ∼ s is ε tot = ε s × ( ε cut ) M × ε acc . (7)Since the cut efficiency and accidentals term are evaluatedseparately for each dataset, the total efficiency term in Eq.7 must be thought of as the signal efficiency for signature s for a specific dataset. A summary of the relevant efficiencyvalues is provided in Table 2, where the per-dataset valuesare exposure weighted over all datasets. The containmentefficiency is the dominant term.3.3 Background ContributionsRadioactive decays and particle interactions other than Tedecay to
Xe excited state, may mimic the process wesearch for. We estimate this background contribution bymeans of background MC simulations described in Sec 3.1.We combine background simulations of different sources,according to the CUORE background model, and from thesimulated background spectra we compute the expected num-ber of background counts for each signature B s , by count-ing the expected events from each source included in thebackground model, and summing the contributions from allsources. We apply the same tight selection cuts around thesignal region defined in Eqs. 5 and 6.We use B s to evaluate an approximate sensitivity for eachsignature and ultimately select the ones that will enter the Table 2
We report the efficiency terms that appear in Eq. 7 separatelyfor the 0 νβ β (top) and 2 νβ β (bottom) analyses. The containment termdominates the efficiency. We report the cut efficiency raised to power M according to the signature it refers to. We quote effective valuescomputed as exposure weighted mean for the cut and accidentals ef-ficiency terms. All values are percentages, the uncertainty on the lastdigit is included in round brackets.0 νβ β Containment 4 . ( ) . ( ) . ( ) Cut 78 . ( ) . ( ) Accidentals 98 . ( ) Total 3 . ( ) . ( ) . ( ) νβ β Containment 4 . ( ) . ( ) . ( ) Cut 78 . ( ) . ( ) Accidentals 98 . ( ) Total 3 . ( ) . ( ) . ( ) analysis (see Sec. 3.4). Once the signatures that enter theanalysis are selected, we optimize the selection cuts aroundthe signal region in order to reject background structureswhile leaving the widest possible sidebands around the ex-pected signal position. In this way we can parameterize thebackground with an appropriate analytical function, whoseshape is dictated by background simulations, and use that toperform the final analysis (see Sec. 4.1). With this methodwe infer the number of reconstructed background events ineach signature from data, rather than relying just on simula-tions.3.4 Experimental Signature RankingThe 15 unique signatures under analysis have different sig-nal efficiencies and backgrounds, and thus different detec-tion sensitivities of the signal. In this section we evaluate anapproximate sensitivity of each signature and reduce the 15signatures down to the most sensitive subset.We analytically evaluate the discovery sensitivity of sig-nature s starting from a background-only model for the totalnumber of counts observed in a single bin centered at the ex-pected signal position. In background-free signatures B s (cid:28) ( µ ) = e − µ where µ is the true value of the number of backgroundcounts. In background-limited ones B s (cid:29) B s . We de-fine the discovery sensitivity as the minimum number ofobserved counts N s such that the probability of observing N > N s counts in the background-only model is smaller thana given threshold p th . Then, from N s , we extract the corre-sponding half life sensitivity˜ S / ( ε s , B s ) = (cid:20) ln ( ) M ∆ t N A η ( Te ) m ( TeO ) (cid:21) S ( ε s , B s ) (8) where M is the detector mass, ∆ t its live time, N A the Avo-gadro constant, η ( Te ) = ( . ± . ) % [46] the iso-topic abundance of Te in natural tellurium, m ( TeO ) = . S ( ε s , B s ) is a score function S ( ε s , B s ) = (cid:40) ε s − ln ( p th ) B s < B th ε s n σ ( p th ) √ B s B s ≥ B th (9)where n σ ( p th ) is the number of Gaussian sigma which cor-respond to p th , and B th sets the transition from thebackground-free approximation to the background-limitedapproximation making S ( ε s , B s ) continuous. For n σ = p th ∼ × − and B th ∼
9. We note though that all signa-tures have a number of expected background counts either < >
10 and their ranking would not be affected by adifferent choice of the B th threshold.We compute the relative score of each signature s as S s . = S ( ε s , B s ) ∑ s (cid:48) S ( ε s (cid:48) , B s (cid:48) ) (10)where s (cid:48) is an index running on all experimental signatures.The efficiency term ε s only includes the MC-based contain-ment efficiency term. This is acceptable for the computationof an approximate analytical score function, since the con-tainment term by far dominates the overall efficiency (Tab.2). We set a threshold of S s >
5% on the score parame-ter S s and we identify three signatures both for the 0 νβ β and 2 νβ β decay search, namely the 2A0-2B1, 2A2-2B3and 3A0 listed in Table 3. The selected experimental sig-natures account for a majority of the sensitivity contribu-tions among studied signatures. The sum of their scores ac-counts for 84% ( ) total score of all signatures in the0 νβ β ( νβ β ) search respectively. We use a phenomenological parameterization of the back-ground in the fitting regions (as opposed to using the pre-dicted spectra from the MC), hence real data are requiredto tune the fit. To avoid biasing our results, we build the fit(Sec. 4.1) on blinded data using the blinding procedure de-scribed in Sec. 4.2.4.1 Fitting TechniqueWe extract the 0 νβ β decay rate and 2 νβ β decay rate of
Te to the 0 + excited state of Xe using two separateBayesian fits. For the single process (0 ν or 2 ν ) the fit is runsimultaneously on all the involved signatures. Every multi-plet of multiplicity M can be represented as a point (cid:126) E ev in a Table 3
Selected experimental signatures for DBD search on the 0 + excited state of Xe in the 0 νβ β (top) and 2 νβ β (bottom) channelare listed. For each signature the corresponding Regions Of Interest(ROI, i.e. the applied selection cuts) are listed in terms of the orderedenergy releases E ≥ E ≥ E . The component that will be used for thefit is highlighted with a ∗ superscript. For each signature the partitionof the secondaries expected to contribute are listed. For each secondarywe report in round brackets the expected energy release in keV. The lastrow reports the corresponding relative score (Eq. 10).0 νβ β E : γ ( ) E : γ ( ) E : γ ( ) E : β β ( ) + γ ( ) E : β β ( ) + γ ( ) E : β β ( ) E : γ ( ) < E < < E ∗ <
573 526 < E < < E ∗ < < E < < E ∗ < < E < νβ β E : β β ( − ) + γ ( ) E : γ ( ) E : β β ( − ) E : γ ( ) E : β β ( − ) + γ ( ) E : γ ( ) E : γ ( ) < E < < E ∗ <
573 400 < E < < E ∗ < < E < < E ∗ < < E + E < M -dimensional space of reconstructed energies. The energyreleases are ordered so that E i > E i + ∀ i = , .., M −
1. Foreach signature, one of the components of the (cid:126) E ev vector isselected to perform the fit. This is referred to as projectedenergy , and indicated with a ∗ superscript in Table 3. In thefollowing we will denote this energy as E ev .An unbinned Bayesian fit is implemented with the BATsoftware package [47]. It allows simultaneous sampling andmaximization of the posterior probability density function(pdf) via Markov Chain Monte Carlo. The likelihood can bedecomposed, for each signature and dataset, as follows:log L s , ds = − ( λ S s , ds + λ B s , ds )++ ∑ ev ∈ ( s , ds ) log (cid:20) λ S s , ds ξ bo , ds f S ( E ev )++ λ B s , ds ξ bo , ds f B ( E ev ) (cid:21) (11)where the subscripts s , bo , ds will be used to refer to a spe-cific signature, bolometer, or dataset respectively. The formof Eq. 11 is the same for 0 νβ β and 2 νβ β , the λ S and λ B terms are the expected number of signal and backgroundevents respectively, ξ bo , ds is the ratio between the exposure of bolometer bo to the exposure of dataset ds , f S and f B arethe normalized signal and background pdfs. They dependjust on the projected energy variable E ev .The response function of CUORE bolometers tomonochromatic energy releases has a functional form de-fined phenomenologically for each bolometer-dataset pair[48] [49] as the superposition of 3 Gaussian componentsto account for non-Gaussian tails. A correction for the biasin the energy scale reconstruction is implemented togetherwith the resolution dependence on energy (see Ref. [21] formore details). The signal term f S ( E ev ) models such shape inthe bolometer-dataset pair the energy E ev was released in.The number of signal counts can be written as λ S s , ds = Γ ( p ) ββ [ yr − ] (cid:20) N A η ( Te ) m ( TeO ) [ g / mol ] (cid:21) ε s ·· ( M ∆ t ) ds [ kg · yr ] ( ε cut ) Mds ( ε acc ) ds (12)where Γ ( p ) ββ is the decay rate of process p and the other pa-rameters were introduced following Eq. 8. The Γ ( p ) ββ param-eter describes the rate of the process p = ν , ν under inves-tigation and is given in both cases a uniform physical prior, Γ ββ > f B ( E ev ) is parameterized as f B ( E ev ) = ∆ E (cid:20) + m s (cid:0) E ev − E ( s ) (cid:1)(cid:21) (13)where ∆ E = E maxs − E mins is the width of the region of inter-est, E ( s ) is the center of it, and m s describes the slope of thebackground for signature s . The normalization of the back-ground term represents the number of expected backgroundcounts λ B s , ds = BI s ( M ∆ t ) ds ( E maxs − E mins ) (14)where BI s is the background index for signature s . Back-ground simulations suggest that a uniform event distribu-tion is enough to describe the continuous background in allsignatures except the 2 νβ β m s parameter is included only when necessary. The back-ground is fully described by the BI s and m s which, together,make 4 (3) nuisance parameters in the 2 νβ β ( νβ β ) caserespectively, that will be marginalized over. The prior forbackground indices BI s and slopes m s is uniform.The combined log-likelihood readslog L ( D | H S + B ) = ∑ s , ds log L s , ds (15)where H S + B indicates that the likelihood is written in thesignal-plus-background model hypothesis H S + B , i.e. that theexistence of the process of interest is assumed.The background-only hypothesis H B is a particular case thatcan be obtained by setting Γ ββ = f S ( E ev | Q s ) centered at the expected position Q s ofthe monochromatic energy release in the projected energyspace.The injection rate of the simulated signal events is com-prised between: Γ minp = · − [ / yr ] and Γ maxp = · − [ / yr ] (16)We then fit the blinded datasets to get data driven esti-mates of the background levels in each fitting window. Thesebackground estimates are used as inputs to our sensitivitystudies in the next section.The results of the fits to the blinded data are reported inTable 4. We see a non-null background for both the2A0 − − background-only Toy Monte Carlo simula-tions (ToyMC), using the numbers in Tab. 4. A background-only ToyMC simulation is an ensemble of simulateddatasets, according to the following procedure which is it-erated N toy times, to produce the same number of ToyMCensembles. We define a set of signatures, together with themultiplicity and cuts in the ordered energy variables thatidentify candidate events. For each signature, we set a func-tional form for the background pdf, either constant or linear,and sample a value from the posterior pdf of the correspond-ing blinded fit. We compute the number of expected back-ground events for each signature and dataset according toEq. 14 and sample the actual number of background eventsfrom a Poisson distribution with expectation value equal tothe number of expected counts.We store each simulated ToyMC event as a vector of or-dered energy releases (cid:126) E ev and related bolometers (cid:126) ch ev , wherethe bolometers are randomly extracted from the activebolometers of each dataset according to their exposure inthe data, while the energies are generated according to theselected shape of the background pdf computed with the pa-rameters (e.g. background index) generated according to theposterior pdfs obtained with the blinded fit to the data.We then fit each ToyMC with the signal-plus-backgroundmodel H S + B and compute the lower limit for the decay half [yr]
90 % C.I. marginalized limit on T2 4 6 8 10 12 14 × × T o y M on t e C a r l o × Expected limit setting sensitivity - Median: 5.58e+24 [yr] MAD: 1.41e+24 [yr] T oy M on t e C a r l o L i m it s [ ]
90 % C . I . marginalized limit on T ν [10 yr]N toy = 10 [yr]
90 % C.I. marginalized limit on T1 2 3 4 5 6 × × T o y M on t e C a r l o × Expected limit setting sensitivity - Median: 2.06e+24 [yr] MAD: 4.92e+23 [yr] T oy M on t e C a r l o L i m it s [ ]
90 % C . I . marginalized limit on T ν [10 yr]N toy = 10 Fig. 2
Distribution of 90% C.I. marginalized upper limits on T / = log2 / Γ for 0 νβ β decay (top) and 2 νβ β decay (bottom) obtained fromToy MC simulations. We obtain a median sensitivity of S ν / = . × yr and S ν / = . × yr (black dashed line), compared to the90% C.I. limit from this analysis (red solid line). life from the 90% quantile of the marginalized posterior pdffor the decay rate parameter. We show the distribution ofsuch limits in Figure 2 for both the 0 νβ β and 2 νβ β de-cay process. We quote the half-life sensitivity as the me-dian limit of the ToyMCs (Table 4). They are respectively:S ν / = ( . ± . ) × yr for the 0 ν decay andS ν / = ( . ± . ) × yr for the 2 ν decay where the un-certainty is the MAD of the corresponding distribution. We show in Figure 3 and Table 5 the results of the fit tounblinded data for both 0 νβ β and 2 νβ β . Though the dataare binned for graphical reasons, the analysis is unbinned.Including contributions from all sources of systematic un-certainty listed in Table 6, no significant signal is observedin either decay mode. The global mode of the joint posterior
Table 4
Results of the blinded fit to 0 νβ β (top) and 2 νβ β (bottom)candidate events in the signatures of Table 3. For each parameter themean and standard deviation of the corresponding marginalized poste-rior distribution are reported. These values are used only as input to thesensitivity studies and fit validation. Final results reported in Table 5.0 νβ β
Observable Blinded Fit Value Units
Blinded Γ νββ . ± . − [ yr − ] BI − . ± . − [ counts / ( keV kg yr )] BI − . ± . − [ counts / ( keV kg yr )] BI . ± . − [ counts / ( keV kg yr )] νβ β Observable Blinded Fit Value Units
Blinded Γ νββ . ± . − [ yr − ] BI − . ± . − [ counts / ( keV kg yr )] m − − . ± . − [ / keV ] BI − . ± . − [ counts / ( keV kg yr )] BI . ± . − [ counts / ( keV kg yr )] Table 5
We report here mean and standard deviation of the marginal-ized posterior distributions for the decay rate and background parame-ters for each signature, derived from unblinded data. The S / param-eter indicates the median expected sensitivity for limit setting at 90%C.I. on the T / parameter together with the MAD of its distribution.The last row reports the marginalized 90% C.I. Bayesian lower limiton the decay half life. All results come from the combined fit withsystematics. 0 νβ β Parameter Final Fit Value Units Γ νββ . ± . − [yr − ]BI − . ± . − [counts/(keV kg yr)]BI − . ± . − [counts/(keV kg yr)]BI . ± . − [counts/(keV kg yr)]S ν / . ± . [yr]T / > . [yr]2 νβ β Parameter Final Fit Value Units Γ νββ . ± . − [yr − ]BI − . ± . − [counts/(keV kg yr)] m − − . ± . − [keV − ]BI − . ± . − [counts/(keV kg yr)]BI . ± . − [counts/(keV kg yr)]S ν / . ± . [yr]T / > . [yr] pdf for the rate parameter isˆ Γ νββ = . + . − . × − yr − (17a)ˆ Γ νββ = . + . − . × − yr − (17b)whereas we quote the uncertainty as the marginalized 68 %smallest interval. We report the marginalized posterior pdfand the 90% C.I. in Figure 4. We include the marginalizedposterior pdfs for individual background parameters in Fig-ure 5 ( νβ β ) and Figure 6 ( νβ β ) . Their means agree with /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/0nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift.rootMon Oct 19 12:19:40 2020 ProjectedEnergy (keV)1250 1255 1260 1265 1270 1275 1280 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)700 710 720 730 740 750 760 C oun t s / . k e V /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/2nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift_linearBkg.rootMon Oct 19 12:23:45 2020 ProjectedEnergy (keV)1220 1230 1240 1250 1260 1270 1280 1290 1300 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/0nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift.rootMon Oct 19 12:19:40 2020 ProjectedEnergy (keV)1250 1255 1260 1265 1270 1275 1280 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)700 710 720 730 740 750 760 C oun t s / . k e V /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/0nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift.rootMon Oct 19 12:19:40 2020 ProjectedEnergy (keV)1250 1255 1260 1265 1270 1275 1280 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)700 710 720 730 740 750 760 C oun t s / . k e V /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/2nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift_linearBkg.rootMon Oct 19 12:23:45 2020 ProjectedEnergy (keV)1220 1230 1240 1250 1260 1270 1280 1290 1300 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V /nfs/cuore1/scratch/gfantini/ROI-excited-files/data/unblinded_5ms1200mm/2nbb/fit_LSfix_highStat/MyModel_ds3519_ds3522_ds3552_ds3555_ds3561_ds3564_ds3567_floatCutEff_floatIsoFrac_floatLSSigma_floatLSBias_floatEffScenario_floatMcEff_floatEffShift_linearBkg.rootMon Oct 19 12:23:45 2020 ProjectedEnergy (keV)1220 1230 1240 1250 1260 1270 1280 1290 1300 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V ProjectedEnergy (keV)525 530 535 540 545 550 555 560 565 570 C oun t s / . k e V Projected Energy [keV] Projected Energy [keV]Projected Energy [keV] Projected Energy [keV]Projected Energy [keV] Projected Energy [keV] C oun t s / . e V C oun t s / . e V C oun t s / . e V C oun t s / . e V C oun t s / . e V C oun t s / . e V (2A0 − 2B1) − 0 νββ (2A0 − 2B1) − 2 νββ (2A2 − 2B3) − 0 νββ (2A2 − 2B3) − 2 νββ (3A0) − 0 νββ (3A0) − 2 νββ Fig. 3
Result of the unbinned fit plotted on binned data. Error barsare just a visual aid and correspond to the square root of the bin con-tents. We show the best fit curve (blue solid), its signal component(blue dashed) at the global mode of the posterior for 0 νβ β (left) and2 νβ β (right), and the 90 % C.I. marginalized limit on the decay rate(red solid). the corresponding results from blinded data within one stan-dard deviation. We observe a slight negative backgroundfluctuation (i.e. limit stronger than expected) in the 0 νβ β decay analysis and a positive one (i.e. limit looser than ex-pected) in the 2 νβ β decay analysis with respect to the me-dian 90% C.I. limit. The probability of setting an evenstronger (looser) limit in the 0 νβ β (2 νβ β ) decay analysisis respectively 45 .
1% and 10 . σ (2 σ ) smallest C.I. of the marginalized pdf We refer here to the Gaussian case to define the probability content ofany n σ interval.0 for the Γ νββ ( Γ νββ ) rate parameter respectively. The follow-ing Bayesian lower bounds on the corresponding half lifeparameters are set: (cid:0) T / (cid:1) ν + > . × yr (
90% C . I . ) (18a) (cid:0) T / (cid:1) ν + > . × yr (
90% C . I . ) (18b)The results reported in this Article represent the moststringent limits on the DBD of Te to excited states andimprove by a factor ∼ Te (seeTable 6). Each systematic uncertainty can be introduced asa set of nuisance parameters in the fit with a specified priordistribution. Each set of nuisance parameters can be acti-vated independently with the priors listed in Table 6. Weindividually monitor the effect of activating each source ofsystematic uncertainty repeating the fit and comparing the90% C.I. Bayesian limit on the half life with respect to the minimal model , where we describe all sources of systematicswith constants rather than fit parameters. Finally, we repeatthe fit activating all additional nuisance parameters at once.We include, for each dataset, two separate parameters tomodel different sources of uncertainty in the cut efficiencyevaluation, and replace the ε cut constant with the sum of ε cut ( I ) + ε cut ( II ) . We refer to cut efficiency I to parameterizethe uncertainty due to the finiteness of the samples of pulserevents and γ decays used to extract the cut efficiency. Itsprior is Gaussian with mean equal to ε cut and width equalto the corresponding uncertainty (see Table. 2). The addi-tional cut efficiency II is uniformly distributed, with zeromean. It models the systematic uncertainty due to the PSAefficiency, shifting the cut efficiency by at most 0 . K peak. We add one nuisance parameter per dataset with aGaussian distributed prior to model this effect. The contain-ment efficiency is instead affected by uncertainty due to thesimulation of Compton scattering events at low energy. Theuncertainty due to the number of simulated signal events isnegligible. We take the ratio of the Compton scattering at-tenuation coefficient to reference data [50] as a measure ofthe relative uncertainty on this efficiency term. We accountfor this, for each signature, introducing a nuisance contain-ment efficiency parameter with a Gaussian prior. The
Tenatural isotopic abundance on natural Te is modeled witha single global nuisance parameter with a Gaussian prior η = ( . ± . ) %. Both the detector response func-tion shape and the energy scale bias are evaluated from data, Source Prior Effect on T / νβ β νβ β Cut efficiency I Gaussian 0 . < . .
1% 0 . . < . .
3% 0 . Te isotopic abundance Gaussian < .
1% 0 . .
3% 0 . .
5% 0 . .
4% 0 . Table 6
Systematic uncertainties. We report each effect separatelyand their combination on the marginalized 90% C.I. T / limit on the0 νβ β and 2 νβ β decay half life. as anticipated in Sec. 4.1. Each effect is separately param-eterized with a 2nd order polynomial as a function of en-ergy, whose coefficients are evaluated with a fit to the 5-7most visible peaks in each dataset. The uncertainty and cor-relations among such parameters are themselves a source ofsystematic uncertainty, and are included as 2 independentsets of correlated parameters per dataset, with a multivariatenormal prior distribution. In this way, the detector responsefunction width and position are allowed to float within theiruncertainty.Uncertainty in modeling the detector response functionleads to the dominant systematic effect on the limit, whichis below a 1% shift. Sub-dominant effects come from theenergy scale bias and the containment efficiency (Table 6). We have presented the latest search for DBD of
Te to thefirst 0 + excited state of Xe with CUORE based on a 372.5kg · yr TeO exposure. We found no evidence for either 0 νβ β nor 2 νβ β decay and we placed a Bayesian lower bound at90% C.I. on the decay half lives of ( T / ) ν + > . × yr for the 0 νβ β mode and ( T / ) ν + > . × yr for the 2 νβ β mode.The median limit setting sensitivity for the 2 νβ β decayof 2 . × yr is starting to approach the 7 . × yr lowerbound of the QRPA theoretical predictions half life range forthis decay mode. The CUORE experiment is steadily tak-ing data at an average rate of ∼
50 kg · yr/month and by theend of its data taking the collected exposure is expected toincrease by an order of magnitude. Work is ongoing to im-prove the sensitivity by extending the analysis to not-fully-contained events, leveraging the information from the topol-ogy of higher dimension coincident signal multiplets to fur-ther reduce the background, and improving the signal effi-ciency by lowering the threshold of the pulse shape discrim-ination algorithm. − × ] -1 [yr ββ Γ × P o s t e r i o r pd f ( a . u . ) Marginalized posterior R - smallest intervals − × ] -1 [yr ββ Γ × P o s t e r i o r pd f ( a . u . ) Marginalized posterior R - smallest intervals M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] Γ νββ [10 −24 yr −1 ]Γ νββ [10 −24 yr −1 ]90 % C . I .90 % C . I . Fig. 4
Marginalized decay rate posterior pdf for 0 νβ β (top) and 2 νβ β (bottom) from the combined fit with all systematics. We show the 90%C.I. in gray.
Acknowledgements
The CUORE Collaboration thanks the directorsand staff of the Laboratori Nazionali del Gran Sasso and the techni-cal staff of our laboratories. This work was supported by the IstitutoNazionale di Fisica Nucleare (INFN); the National Science Foundationunder Grant Nos. NSF-PHY-0605119, NSF-PHY-0500337, NSF-PHY-0855314, NSF-PHY-0902171, NSF-PHY-0969852, NSF-PHY-1307204,NSF-PHY-1314881, NSF-PHY-1401832, and NSF-PHY-1913374; andYale University. This material is also based upon work supported bythe US Department of Energy (DOE) Office of Science under Con-tract Nos. DE-AC02-05CH11231 and DE-AC52-07NA27344; by theDOE Office of Science, Office of Nuclear Physics under Contract Nos.DE-FG02-08ER41551, DE-FG03-00ER41138, DE-SC0012654, DE-SC0020423, DE-SC0019316; and by the EU Horizon2020 researchand innovation program under the Marie Sklodowska-Curie Grant Agree-ment No. 754496. This research used resources of the National EnergyResearch Scientific Computing Center (NERSC). This work makes useof both the DIANA data analysis and APOLLO data acquisition soft-ware packages, which were developed by the CUORICINO, CUORE,LUCIFER and CUPID-0 Collaborations.
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The CUORE experiment: from the commis-sioning to the first νβ β limit . PhD thesis, Universitàdegli studi di Genova, 2018.50. J. Allison et al. Recent developments in Geant4. Nucl.Instrum. Meth. A , 835:186–225, 2016. − × [cts/keV/kg/yr] BI00.511.522.5 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_0 - smallest intervals − × [cts/keV/kg/yr] BI00.20.40.60.811.21.41.61.8 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_1 - smallest intervals − × [cts/keV/kg/yr] BI012345 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_2 - smallest intervals M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] BI [10 −3 counts/(keV kg yr)]BI [10 −3 counts/(keV kg yr)]BI [10 −3 counts/(keV kg yr)]99.7 % C . I .95.5 % C . I .68.3 % C . I .99.7 % C . I .95.5 % C . I .68.3 % C . I .99.7 % C . I .95.5 % C . I .68.3 % C . I . Fig. 5
Marginalized posterior pdf for the background parameters ofthe 0 νβ β model from the combined fit with all systematics. We showthe 68 . , . , .
7% smallest C.I. in green, yellow and red respec-tively. − × [cts/keV/kg/yr] BI00.20.40.60.811.21.41.61.8 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_0 - smallest intervals − × [cts/keV/kg/yr] BI00.20.40.60.811.21.41.6 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_1 - smallest intervals − × [cts/keV/kg/yr] BI012345 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BI_2 - smallest intervals − − − − − × [cts] m00.20.40.60.811.21.41.6 × P o s t e r i o r pd f ( a . u . ) Marginalized posterior BIslope_0 - smallest intervals M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] M a r g i n a li ze dpo s t e r i o r pd f[ a . u . ] BI [10 −3 counts/(keV kg yr)]BI [10 −3 counts/(keV kg yr)]BI [10 −3 counts/(keV kg yr)] m [10 −3 keV −1 ]99.7 % C . I .95.5 % C . I .68.3 % C . I .99.7 % C . I .95.5 % C . I .68.3 % C . I .99.7 % C . I .95.5 % C . I .68.3 % C . I .99.7 % C . I .95.5 % C . I .68.3 % C . I . Fig. 6
Marginalized posterior pdf for the background parameters ofthe 2 νβ β model from the combined fit with all systematics. We showthe 68 . , . , ..